FRACTAL SPACE-TIME AND THE STATISTICAL-MECHANICS OF RANDOM-WALKS

Authors
Citation
Gn. Ord, FRACTAL SPACE-TIME AND THE STATISTICAL-MECHANICS OF RANDOM-WALKS, Chaos, solitons and fractals, 7(6), 1996, pp. 821-843
Citations number
29
Categorie Soggetti
Mathematics,Mechanics,Engineering,"Physics, Applied
ISSN journal
09600779
Volume
7
Issue
6
Year of publication
1996
Pages
821 - 843
Database
ISI
SICI code
0960-0779(1996)7:6<821:FSATSO>2.0.ZU;2-N
Abstract
We illustrate some of the ideas involved in fractal space-time using f amiliar deterministic fractals. Starting with the objective of reprodu cing the Heisenburg uncertainty principle for point particles, we use the Peano-Moore curve to help visualize the qualitative behaviour of p articles moving on fractal trajectories in space and time. With this q ualitative picture in mind we then explore exactly solvable models to verify that our ideas are mathematically consistent. We find that the Schrodinger equation describes ensembles of classical particles moving on fractal random walk trajectories. This shows that the Schrodinger equation has a straightforward microscopic model which is not, however , appropriate for quantum mechanics. The free particle Dirac equation is also derivable in terms of ensembles of classical particles and thi s unites the two equations conceptually in a very direct way. In both cases what we discover is a many-particle simulation of quantum mechan ics and this confirms in a graphic way that the mysteries surrounding quantum mechanics lie not in the equations, but in interpretation and the theory of measurement. Finally, we discuss an exactly solvable mod el which incorporates fractal time. The calculation produces the Dirac equation in 1 + 1 dimensions and because of intrinsic space-time loop s, constitutes a model with the potential to exhibit the wave-particle duality found in nature. Copyright (C) 1996 Elsevier Science Ltd